2. If I am to pick 3 random letters from a bag with the letters [tex]$\{\underline{A}, B, \underline{C}, \underline{D}\}$[/tex] given that each letter picked is permanently removed from the bag, what is the probability that my pick follows the order ADC?

A. [tex]$\frac{1}{24}$[/tex]
B. [tex]$\frac{3}{16}$[/tex]
C. [tex]$\frac{1}{32}$[/tex]
D. [tex]$\frac{1}{12}$[/tex]



Answer :

Let's tackle this problem step-by-step, focusing on finding the probability of picking the letters A, D, and C in that specific order from a set of given letters.

First, note that the letters present in the bag are A, B, C, and D. The question requires us to pick 3 letters out of these 4, in a specific order: A, D, and C.

### Step 1: Determine the Total Number of Permutations

To determine the total number of ways in which we can pick 3 letters out of 4 in any order, we use permutations [tex]\( P(n, k) \)[/tex], where [tex]\( n \)[/tex] is the total number of items to choose from (4 letters), and [tex]\( k \)[/tex] is the number of items to pick (3 letters).

[tex]\[ P(4, 3) = \frac{4!}{(4-3)!} = \frac{4!}{1!} = 4 \times 3 \times 2 = 24 \][/tex]

This means there are 24 possible ways to arrange 4 letters taking 3 at a time.

### Step 2: Determine the Number of Favorable Outcomes

The question asks us to find the probability of picking the letters in a specific order: A, D, and C. For a specific order of three letters, there is only 1 favorable outcome (ADC).

### Step 3: Calculate the Probability

The probability [tex]\( P(E) \)[/tex] of an event [tex]\( E \)[/tex] is given by the ratio of the number of favorable outcomes to the total number of possible outcomes.

[tex]\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of permutations}} \][/tex]

Substituting the values,

[tex]\[ P(\text{Picking ADC}) = \frac{1}{24} \][/tex]

### Final Answer

Thus, the probability that the letters picked in order are A, D, and C is:

[tex]\[ \boxed{\frac{1}{24}} \][/tex]

This corresponds to option A.